Problems related to the idea of infinity are among the most fundamental and have attracted the attention of the most brilliant thinkers throughout the whole history of humanity. Numerous trials have been done in order to evolve existing numeral systems and to include infinite and infinitesimal numbers in them. Such eminent researchers as Aristotle, Archimedes, Euclid, Eudoxus, Parmenides, Plato, Pythagoras, Zeno, Cantor, Dedekind, Descartes, Leibniz, Newton, Peano, Cohen, Frege, Gelfond, Gödel, Robinson, and Hilbert worked hard on these topics. To emphasize importance of the subject it is sufficient to mention that the Continuum Hypothesis related to infinity has been included by David Hilbert as the problem number one in his famous list of 23 unsolved mathematical problems that have influenced strongly development of the mathematics in the 20th century.
The point of view on infinity accepted nowadays is based on the ideas of George Cantor who has shown that there exist infinite sets having different number of elements. Particularly, he has shown that the infinite set of natural numbers, N, has less elements than the set, R, of real numbers. There exist different ways to generalize arithmetic for finite numbers to the case of infinite numbers. However, arithmetics developed for infinite numbers are quite different with respect to the arithmetic we are used to deal with (see examples in [1]). Moreover, very often they leave undetermined many operations where infinite numbers take part (for example, infinity minus infinity, infinity divided by infinity, sum of infinitely many items, etc.) or use representation of infinite numbers based on infinite sequences of finite numbers. These crucial difficulties did not allow to people to construct computers that would be able to work with infinite and infinitesimal numbers in the same manner as we are used to do with finite real numbers.
In fact, in modern computers, only arithmetical operations with finite numbers or with intervals having finite numbers as their limits (see, for example [2]) are realized. Traditional real numbers can be represented in computer systems in various ways. Many of them use positional numeral systems with the finite radix b. Note that numeral is a symbol or group of symbols that represents a number. The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘3’, ‘three’, and ‘III’ are different numerals, but they all represent the same number.
In positional numeral systems fractional numbers are expressed by the record(anan−1 . . . a1a0.a−1a−2 . . . a−(q−1)a−q)b  (1)where numerals ai, −q≦i≦n, are called digits, belong to the alphabet {0, 1, . . . , b−1}, and the dot is used to separate the fractional part from the integer one. Thus, the numeral (1) is equal to the sumanbn+an−1bn−1+ . . . +a1b1+a0b0+a−1b−1+ . . . +a−(q−1)b−(q−1)+a−qb−q  (2)
In modern computers, the radix b=2 with the alphabet {0, 1} is mainly used to represent numbers. There exist numerous ways to represent and to store numbers in computers. Particularly, a floating-point representation expresses a number in four parts: a sign, a mantissa, a radix, and an exponent. The sign is either a 1 or −1. The mantissa, always a positive number, holds the significant digits of the floating-point number. The exponent indicates the positive or negative power of the radix that the mantissa and sign should be multiplied by.